Computation of the effective slip of rough hydrophobic surfaces via homogenization

We present a quantitative analysis of the effect of rough hydrophobic surfaces on viscous Newtonian flows. We use a model introduced by Ybert and coauthors in [Achieving large with superhydrophobic surfaces: Scaling laws for generic geometries, Phys. Fluids 19 (2007) 123601], in which the rough surface is replaced by a flat plane with alternating small areas of slip and no-slip. We investigate the averaged slip generated at the boundary, depending on the ratio between these areas. This problem reduces to the homogenization of a nonlocal system, involving the Dirichlet to Neumann map of the Stokes operator, in a domain with small holes. Pondering on the works of Allaire [Homogenization of the Navier-Stokes equations in open sets perforated with tiny holes. I. Abstract framework, a volume distribution of holes, Arch. Rational Mech. Anal. 113 (1990) 209-259; Homogenization of the Navier-Stokes equations in open sets perforated with tiny holes. II. Noncritical sizes of the holes for a volume distribution and a surface distribution of holes, Arch. Rational Mech. Anal. 113 (1990) 261-298]. We compute accurate scaling laws of the averaged slip for various types of roughness (riblets, patches). Numerical computations complete and confirm the analysis.